Hecke-Operator

In mathematics we mean by Hecke operators certain linear operators on the vector space of all modular forms. Introduced these operators by Erich Hecke. Their meaning is obtained by certain modular forms are simultaneous eigenfunctions for all Hecke operators and can thus be drawn conclusions about the properties of the Fourier coefficients of these functions. These modular forms are also called eigenmodes.

Definition

It should be the vector space of all modular forms k to the weight

A Hecke operator is a linear map

For primes p this reduces to

Properties and Applications

The Hecke operators form into itself, that is, is again an entire modular form of weight k, in particular, they form cusp forms, ie Modular forms with a zero at, again from tip shapes.

Has a Fourier expansion,

Thus, a Fourier expansion

We call the function f is a simultaneous eigenform if f eigenform for all Hecke operators is, in this case, the eigenvalues

The vector space of cusp forms even has a base of simultaneous eigenfunctions for the operator, thus results, for example for the discriminant, the only up to a constant factor peak form of weight 12:

And its Fourier coefficients Ramanujan tau function applies:

Specifically, n is relatively prime m, then, that is, the number-theoretic function is multiplicative.

The only non-peak shapes that are simultaneous eigenforms for all Hecke operators, are the normalized Eisenstein series

For the Fourier coefficients of the Eisenstein series yields:

And for relatively prime m, n this reduces to back, ie also the number-theoretic function is multiplicative.